Poor John returned to his cellar, taking the three men with him, and then his task began to perplex him. Of full bottles he had seven large and seven small, and of empty bottles five large and five small, as shown in the ill.u.s.tration. How was he to make the required equitable division?

He divided the bottles into three groups in several ways that at first sight seemed to be quite fair, since two small bottles held just the same quant.i.ty of wine as one large one. But the large bottles themselves, when empty, were not worth two small ones.

Hence the abbot's order that each man must take away the same number of bottles of each size.

Finally, the cellarman had to consult one of the monks who was good at puzzles of this kind, and who showed him how the thing was done. Can you find out just how the distribution was made?

77.--_Making a Flag._

[Ill.u.s.tration]

A good dissection puzzle in so few as two pieces is rather a rarity, so perhaps the reader will be interested in the following. The diagram represents a piece of bunting, and it is required to cut it into two pieces (without any waste) that will fit together and form a perfectly square flag, with the four roses symmetrically placed. This would be easy enough if it were not for the four roses, as we should merely have to cut from A to B, and insert the piece at the bottom of the flag. But we are not allowed to cut through any of the roses, and therein lies the difficulty of the puzzle. Of course we make no allowance for "turnings."

78.--_Catching the Hogs._

In the ill.u.s.tration Hendrick and Katrun are seen engaged in the exhilarating sport of attempting the capture of a couple of hogs.

Why did they fail?

[Ill.u.s.tration]

Strange as it may seem, a complete answer is afforded in the little puzzle game that I will now explain.

Copy the simple diagram on a conveniently large sheet of cardboard or paper, and use four marked counters to represent the Dutchman, his wife, and the two hogs.

At the beginning of the game these must be placed on the squares on which they are shown. One player represents Hendrick and Katrun, and the other the hogs. The first player moves the Dutchman and his wife one square each in any direction (but not diagonally), and then the second player moves both pigs one square each (not diagonally); and so on, in turns, until Hendrick catches one hog and Katrun the other.

This you will find would be absurdly easy if the hogs moved first, but this is just what Dutch pigs will not do.

79.--_The Thirty-one Game._

This is a game that used to be (and may be to this day, for aught I know) a favourite means of swindling employed by card-sharpers at racecourses and in railway carriages.

As, on its own merits, however, the game is particularly interesting, I will make no apology for presenting it to my readers.

The cardsharper lays down the twenty-four cards shown in the ill.u.s.tration, and invites the innocent wayfarer to try his luck or skill by seeing which of them can first score thirty-one, or drive his opponent beyond, in the following manner:--

One player turns down a card, say a 2, and counts "two"; the second player turns down a card, say a 5, and, adding this to the score, counts "seven"; the first player turns down another card, say a 1, and counts "eight"; and so the play proceeds alternately until one of them scores the "thirty-one," and so wins.

Now, the question is, in order to win, should you turn down the first card, or courteously request your opponent to do so? And how should you conduct your play? The reader will perhaps say: "Oh, that is easy enough.

You must play first, and turn down a 3; then, whatever your opponent does, he cannot stop your making ten, or stop your making seventeen, twenty-four, and the winning thirty-one. You have only to secure these numbers to win."

But this is just that little knowledge which is such a dangerous thing, and it places you in the hands of the sharper.

You play 3, and the sharper plays 4 and counts "seven"; you play 3 and count "ten"; the sharper turns down 3 and scores "thirteen"; you play 4 and count "seventeen"; the sharper plays a 4 and counts "twenty-one"; you play 3 and make your "twenty-four."

[Ill.u.s.tration:

[1] [1] [1] [1]

[2] [2] [2] [2]

[3] [3] [3] [3]

[4] [4] [4] [4]

[5] [5] [5] [5]

[6] [6] [6] [6] ]

Now the sharper plays the last 4 and scores "twenty-eight." You look in vain for another 3 with which to win, for they are all turned down! So you are compelled either to let him make the "thirty-one" or to go yourself beyond, and so lose the game.

You thus see that your method of certainly winning breaks down utterly, by what may be called the "method of exhaustion." I will give the key to the game, showing how you may always win; but I will not here say whether you must play first or second: you may like to find it out for yourself.

80.--_The Chinese Railways._

[Ill.u.s.tration]

Our ill.u.s.tration shows the plan of a Chinese city protected by pentagonal fortifications. Five European Powers were scheming and clamouring for a concession to run a railway to the place; and at last one of the Emperor's more brilliant advisers said, "Let every one of them have a concession!" So the Celestial Government officials were kept busy arranging the details. The letters in the diagram show the different nationalities, and indicate not only just where each line must enter the city, but also where the station belonging to that line must be located.

As it was agreed that the line of one company must never cross the line of another, the representatives of the various countries concerned were engaged so many weeks in trying to find a solution to the problem, that in the meantime a change in the Chinese Government was brought about, and the whole scheme fell through. Take your pencil and trace out the route for the line A to A, B to B, C to C, and so on, without ever allowing one line to cross another or pa.s.s through another company's station.

81.--_The Eight Clowns._

[Ill.u.s.tration]

This ill.u.s.tration represents a troupe of clowns I once saw on the Continent. Each clown bore one of the numbers 1 to 9 on his body. After going through the usual tumbling, juggling, and other antics, they generally concluded with a few curious little numerical tricks, one of which was the rapid formation of a number of magic squares. It occurred to me that if clown No. 1 failed to appear (as happens in the ill.u.s.tration), this last item of their performance might not be so easy.

The reader is asked to discover how these eight clowns may arrange themselves in the form of a square (one place being vacant), so that every one of the three columns, three rows, and each of the two diagonals shall add up the same. The vacant place may be at any part of the square, but it is No. 1 that must be absent.

82.--_The Wizard's Arithmetic._